Execute a specific action on the first run of the script?
Use the window manager

Window manager properties only exist for a session, whereas scene properties are stored in the file.

It's good practice to avoid try except clauses. If a custom property "xxx" on object ob is defined, it will be a member of ob.keys() and can be tested for with "xxx" in ob.keys()

import bpy
context = bpy.context
wm = context.window_manager

if "new" in wm.keys():
    print("Not New")
    wm["new"] += 1
else:
    wm["new"] = 1
    print("New")


May also be an idea to investigate using a non persistent  on-load handler. (Remove the decorator from the example)